The movie Black Robe depicts French and Native American interaction in the Great Lakes region. In it, the Algonquin society is portrayed as more egalitarian than the French: Algonquin women are not under close sexual control, they appear to have a kind of social equality so that they make speak their minds freely, and they are not obviously discriminated against. The French, on the other hand, have no women in their settlement, and indeed seem to hardly have women at all, only a few appearing in one character's dreams.
Despite the superficial equality shown in Algonquin women, however, the movie depicts a more implicit patriarchy: the men make all the important decisions, the men are the only ones who deal with the French in political decisions, and the men seem to be the real leaders of the groups.
Is this accurate? What were gender relations like among the 17th-century Algonquin, or other culturally similar groups?
Though the characters are fictional, Black Robe is based on a novel by the same name, which in turn is based on the Jesuit Relations. Unfortunately, the greatly abridged copy of the Relations that I have on hand at the moment, focuses mainly on the Innu and the Wendat (the Montagnais and the Huron, respectively) rather than the Algonquin. The Innu are close cultural cousins of the Algonquin, so we'll have to file this answer under "culturally similar group." (It helps that some of the incidents incorporated in the film are actually based on events that happened with Jesuits among the Innu--such as Paul le Jeune's poorly planned winter journey)
A major issue in this film is that we primarily see the Native characters in a limited number of circumstances. Namely, the Algonquins are traveling between two foreign powers--the French colony at Quebec and the Wendat villages in Wendake. Trade and travel were arenas where men served as spokespersons and leaders. The women likely would be consulted in private, but since the French weren't bring any women to any trade negotiations, it would been seen as improper for women to speak at a de facto men's council.
In the film we didn't get a chance to see the arenas where women would have held power (or rather the fact that women were making the decisions wasn't something highlighted in the film). Women choose where specificallly to build campsites (though one of the hunters / men usually led the way to a general area were hunting seemed promising) and how to distribute the products of the band's labor. Le Jeune recounts a particular incident in which an Innu woman was being less than frugal with the band's reserves of food, even then none of the men argued with her about her potentially ill-advised (in le Jeune's estimation) generosity. It was her right to distribute the food as she pleased, and it was not the men's place countermand her.